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Problem 11

Denote by \(\mathcal{K}\) the class of functions $[0, \infty) \rightarrow[0, \infty)$ which are zero at zero, strictly increasing, and continuous, by \(\mathcal{K}_{\infty}\) the set of unbounded \(\mathcal{K}\) functions, and by \(\mathcal{K} \mathcal{L}\) the class of functions $[0, \infty)^{2} \rightarrow[0, \infty)\( which are of class \)\mathcal{K}$ on the first argument and decrease to zero on the second argument. A continuous-time time-invariant system \(\dot{x}=f(x, u)\) with state space \(X=\mathbb{R}^{n}\) and control-value space \(\mathcal{U}=\mathbb{R}^{m}\) is said to be input-to-state stable (ISS) if there exist a function \(\beta \in \mathcal{K} \mathcal{L}\) and a function \(\gamma \in \mathcal{K}_{\infty}\) so that the following property holds: For each \(T>0\), if \(\omega\) is a control on the interval \([0, T]\) and \(\xi\) is a solution of \(\dot{\xi}=f(\xi, \omega)\) on the interval \([0, T]\), then $$ \|\xi(T)\| \leq \max \left\\{\beta(\|\xi(0)\|, T), \gamma\left(\|\omega\|_{\infty}\right)\right\\} $$

Expert verified

To summarize, a continuous-time, time-invariant system \(\dot{x}=f(x, u)\) with state space \(X=\mathbb{R}^{n}\) and control-value space \(\mathcal{U}=\mathbb{R}^{m}\) is Input-to-State Stable (ISS) if there exist functions \(\beta \in \mathcal{K} \mathcal{L}\) and \(\gamma \in \mathcal{K}_{\infty}\) such that the following property holds for any given \(T>0\), control function \(\omega\) on the interval \([0, T]\), and solution \(\xi\) of the differential equation \(\dot{\xi}=f(\xi, \omega)\) on interval \([0, T]\):
\(\|\xi(T)\| \leq \max \left\{\beta(\|\xi(0)\|, T), \gamma\left(\|\omega\|_{\infty}\right)\right\}\)

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Chapter 7

The time-invariant complete system \(\Sigma\) is asymptotically observable (with respect to \(x^{0}, u^{0}, y^{0}\) ) if \(\Sigma^{0}\) is globally asymptotically stable (with respect to \(x^{0}\) ).

Chapter 7

Let \(\Sigma_{1}\) be a topological system, and pick any \(x_{1}^{0} \in X\). The system \(\Sigma_{1}\) is dynamically stabilizable (with respect to \(x_{1}^{0}\) ) by \(\left(\Sigma_{2}, x_{2}^{0}, k_{1}, k_{2}\right)\), where \(\Sigma_{2}\) is a topological system, if the interconnection of \(\Sigma_{1}\) and \(\Sigma_{2}\) through \(k_{1}, k_{2}\) is topologically well-posed and is globally asymptotically stable with respect to \(\left(x_{1}^{0}, x_{2}^{0}\right)\).

Chapter 7

When the system \(\Sigma\) is discrete-time and observable, one may pick a matrix \(L\) so that \(A+L C\) is nilpotent, by the Pole-Shifting Theorem. Thus, in that case a deadbeat observer results: The estimate becomes exactly equal to the state after \(n\) steps.

Chapter 7

The behavior \(\Lambda\) is UBIBO if and only if its impulse response \(K\) is integrable.

Chapter 7

For linear time-invariant continuous-time or discrete-time systems, \((A, C)\) is asymptotically observable if and only if $\left(A^{\prime}, C^{\prime}\right)$ is asymptotically controllable.

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